Problem 80 - Algebra 1: Concepts and Skills

Question

Solve the linear system. (Lessons 7.2,7.3) $$ \begin{array}{r} {y=4 x} \\ {x+y=10} \end{array} $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 2\) and \(y = 8\).

Step 1: Substitute y in the second equation

We substitute \(y\) from the first equation, \(y=4x\), into the second equation. This gives us: \(x + 4x = 10\)

Step 2: Solve for x

Add the like terms resulting in: \(5x=10\). Then, divide both sides by 5 to solve for \(x\), yielding: \(x=2\)

Step 3: Substitute x in the first equation

We substitute \(x=2\) into the first equation \(y=4x\) to solve for \(y\), giving \(y = 4 * 2\)

Step 4: Solve for y

This equation simplifies to \(y = 8\)

Key Concepts

Substitution Method

The substitution method is a powerful tool used in algebra for solving systems of linear equations. It involves replacing one variable with an equivalent expression from another equation. This method works particularly well when one equation in the system is already solved for a specific variable.

To use substitution effectively, follow these steps:

• Identify an equation where a variable is isolated, such as \( y = 4x \).
• Substitute this expression into the other equation(s) in the system, replacing the isolated variable.
• Simplify the resulting equation and solve for the remaining variable.
• Plug this solution back into one of the original equations to find the value of the isolated variable.

By consistently applying the substitution method, students can untangle complex systems of equations, one variable at a time.

Linear Equations

Linear equations form the foundation of algebra and represent relationships between variables with straight-line graphs. A linear equation in two variables, like \( x \) and \( y \), generally takes the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.

Characteristics of linear equations include:

• The exponents on the variables are always 1.
• The graph of a linear equation is always a straight line.
• Each linear equation in a system can usually be manipulated to isolate one variable or another.

Familiarity with linear equations enables students to solve problems involving rates of change, among various other applications.

Solving for Variables

Solving for variables is the process of determining the value of variables that make a given equation true. In the context of a system of equations, once a variable is isolated, as in \( y = 4x \), it becomes simpler to find the values of both variables.

General steps to solve for variables include:

• Isolating the variable on one side of the equation, leaving a constant or an expression without the variable on the other side.
• Using basic arithmetic operations to simplify the equation and solve for the variable.
• Checking the solution by substituting the values back into the original equations.

It's essential to perform the same operation on both sides of the equation to maintain equality, which leads to finding accurate solutions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations without an equality sign. They can be as simple as \( 4x \) or more complex with multiple terms and operations.

Understanding algebraic expressions is key to solving equations because:

• They can be manipulated through the application of algebraic rules.
• It's important to know how to combine like terms, such as adding \( x \) and \( 4x \) to make \( 5x \), which is a fundamental step in simplification.
• Recognizing and factoring out common factors in expressions can help simplify solving equations.

Through the mastery of algebraic expressions, students are able to translate verbal statements into mathematical terms, which is an invaluable skill in problem-solving.